Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/4` and the angle between sides B and C is `pi/12`. If side B has a length of 42, what is the area of the triangle?

Answer 1

Area of `Delta = 187.5281`

Explanation:

`/_ A = pi/12, /_ C = pi/4, /_ B = (pi - ((pi/12) + (pi/4))) = (2pi)/3`

`A / sin A = B / sin B = C / sin C`

`A / sin (pi/12) = 42/ sin ((2pi)/3) = C / sin (pi/4)`

`A = ( 42 * sin (pi/12)) / sin ((2pi)/3) = (42 * 9.2588)/0.866 = 12.5515`

`C = (42 * sin (pi/4)) / sin ((2pi)/3) = (42 * 0.707)/0.866 = 34.2887`

Semi perimeter `s = (A + B + C)/2 = (12.5515 + 42 + 34.2887)/2`
`s = 44.4201`

`s-A = 44.4201-12.5515 = 31.8686`
`s-B = 44.4201-42 = 2.4201`
`s-C = 44.4201-34.2887 = 10.1314`

Area of `Delta = sqrt(s (s-A) (s-B) (s-C))`
Area of `Delta = sqrt(44.4201 * 31.8686 * 2.4201 * 10.1314)`
Area of `Delta = sqrt(35166.8011) = 187.5281`